The redshift observation of distant type 1A Supernovae in 1998 is interpreted as cosmological redshift from the expanding universe and reopened the debate of the need for a cosmological constant (dark energy) in our cosmological model as the simplest explanation for the accelerated expansion of the universe. This cosmological constant can be thought as a constant of curvature throughout all space or equivalently, a constant energy density of the vacuum, which drives acceleration expansion by exerting negative pressure. The simplest solution to explain the accelerated expansion is to hypothesize a constant energy field exerting negative pressure, which is referred to as dark energy in the Lambda-CDM model.
The expansion of the universe not only dilutes radiation and matter energy densities but also for the case of radiation (photons, relativistic neutrinos, or gravitational waves), their energy is lost through redshift due to the fact that it is inversely proportional to their wavelength, which is stretched by the expansion of space: Energy of a photon = Planck’s constant * speed of light / wavelength.
The current value for the photon energy density comes from the Cosmic Microwave Background (CMB), emitted at the recombination epoch (the first 370.000 years of the universe). Astrophysical sources of photon energy density (for instance, stars and dust emission) can be neglected against the CMB because their number and energy are estimated to be at least two and one order of magnitude smaller, respectively.
In contrast, relic relativistic massless neutrinos were created 1 second after the Big Bang. At the start of the matter-dominated era (the first 60.000 years of the universe), neutrinos decoupled from other matter becoming nonrelativistic, behaving in terms of dilution as regular matter without redshifting.
Gravitational waves propagate through space at the speed of light, and there is now evidence for a stochastic gravitational wave background, composed of many localized, unresolved, and independent gravitational waves from different sources, which can be thought of as the accumulation of these waves spread across spacetime. Gravitational waves are barely absorbed nor reflected to any significant degree, so the dissipation of their energy takes place predominantly in redshift. Its total energy density is unknown.
As opposed to radiation, the vacuum energy or cosmological constant does not dilute with the expansion of the universe, as it is a constant value of energy density. Thus, total cosmological constant energy (not density) increases proportionally to the volume as the universe expands, and energy is gained.
This loss and gain of energy is allowed to take place since global energy conservation cannot be defined in General Relativity because there is no time translation invariance in the expansion of the universe.
The energy lost in CMB redshift per unit of volume is just an order of magnitude smaller than the energy gained by the cosmological constant (the energy density of dark energy) per unit of volume since the recombination epoch. This suggests that energy conservation could be imposed by accounting for more contributions to the CMB lost energy, such as gravitational wave redshift, to match the energy gained by the cosmological constant.
To apply energy conservation, first, the energy that has been gained by the cosmological constant along the universe scale factor per unit of space volume must be estimated. The cosmic scale factor is used to characterize the expansion of the universe (representing the relative size of the universe at a given time compared to its size at present epoch). This must be the energy lost by gravitational waves (plus photons) redshift along the scale factor. The cosmological constant energy density at the early universe can be neglected and considered from recombination epoch. Then, the energy density of the gravitational wave background, so that its redshifted energy lost is equal to the gained cosmological constant energy at any given scale factor can be obtained, accounting for its dilution. Additionally, the rate of gravitational wave background energy generation from astrophysical sources through the scale factor can be calculated. Finally, the values of the energy density of the gravitational wave background along the scale factor and its nowadays value can be derived.
The main issue for the energy gained by the cosmological constant to be equal to the energy lost by photon and gravitational wave redshifts is that both energy densities lose energy slowly while the energy gained by the cosmological constant occurs quicker, in terms of scale factor. Both can only be equaled if most of the gravitational wave energy density is produced along the universe’s age, for instance, by astrophysical sources. Thus, the gravitational wave energy density cannot be simply calculated from current value of gravitational wave energy density and the size of the universe through the scale factor.
Current progress in gravitational wave detection (such as LIGO/VIRGO and future LISA detectors) suggests that the energy density of the stochastic gravitational wave background will be estimated in the next decades.
In this proposal, the cosmological constant field exchanges energy with the electromagnetic and gravitational field, which is natural since the electromagnetic field is a source of energy and thus, a source of gravity in General Relativity. Energy conservation might be another condition to be imposed to General Relativity to properly describe physical reality, together with energy conditions.
One could argue that massive particles should not contribute significantly to the energy loss that is transferred to the cosmological constant as dimensionally altered by the expansion because their interactions reset the difference in distances, although they are certainly affected in some way (they are described by wave functions subject to redshift in the standard interpretation of quantum mechanics). Also, virtual massless particles should not in principle be affected either. If these values are significant enough, the estimated gravitational wave energy density would be smaller. The same would happen for other unknown contributions, such as particles decaying into vacuum energy.
Assuming that the cosmological constant energy density is constant thought-out space, the hidden underlying mechanism for energy conservation through redshift must be non-local, hinting that the mechanism has a quantum nature (the process of wave function collapse in the standard quantum mechanics interpretation is non-local). If the cosmological constant is not constant through space, locality may be preserved and regions with greater amount of it would imply greater past gravitational wave energy density. If the cosmological constant is not constant through time, the Hubble tension could be resolved and the age of the universe estimations would change. Also, a different fate for the universe instead of the big freeze could occur.
Cosmic inflation could also be described by the same transition of energy due to redshift to a quantum field such as the scalar inflaton field.
Would an estimation of the energy density of gravitational waves based on the required redshift in order to account for dark energy with the right amount of production of gravitational waves along the universe scale factor, match the future estimations that will be perfomed by the upcoming laser interferometers of low frequency gravitational waves? Could this mechanism of energy conservation be implemented into the first Friedmann equation? Is energy conserved through a non-local quantum mechanical process related to dynamical spacetime?
Research published in Advances in Astronomy, vol. 2023, Article ID 2882534, 2023
Available at: Researchgate